[seqfan] Re: The Sequencer OEIS survey

[Max Alekseyev]

The formula for A210379 (Number of 2×2 matrices with all terms in 0,1,…,n
and odd trace) is given with wrong offset.
It should be a(n) = (n+1)^2 * floor( (n+1)^2/2 ). This formula is easy to
prove.

Namely, the factor (n+1)^2 stands for the number of elements at entries
(1,2) and (2,1) of the matrix, which do not participate in the trace
calculation and thus each can independently take all n+1 different values.
The factor floor( (n+1)^2/2 ) counts number of pairs of elements (indexed
(1,1) and (2,2) in the matrix) from {0,1,...,n} with odd sum. If n is odd,
there are (n+1)/2 even and (n+1)/2 odd numbers in the set, implying that
the number of pairs with odd sum is (n+1)/2*(n+1)/2 + (n+1)/2*(n+1)/2 =
(n+1)^2/2 = floor( (n+1)^2/2 ). If n is even, there are (n+2)/2 even and
n/2 odd numbers in the set, implying that there are (n+2)/2*n/2 +
n/2*(n+2)/2 = n*(n+2)/2 = floor( (n+1)^2/2 ).

Regards,
Max



On Sun, Apr 12, 2015 at 6:28 AM, Philipp Emanuel Weidmann <
pew at worldwidemann.com> wrote:

> Back in February I announced a project to scan all OEIS sequences using
> the Sequencer system (https://github.com/p-e-w/sequencer) in order to
> identify new closed-form expressions for the sequence terms.
>
> The search has now concluded and the results are available at
> http://worldwidemann.com/the-sequencer-oeis-survey/
>
> Please note that only sequences without a "formula" field were scanned
> as explained in the article.
>
> Best regards
> Philipp Emanuel Weidmann
>
>
>
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